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A new two-dimensional model of rolling-sliding contact creep curves
for a range of lubrication types
DI Fletcher*
Department of Mechanical Engineering, University of Sheffield, Mappin Street,
Sheffield, UK S1 3JD
* Corresponding author, Tel +44 114 2227760, Fax +44 114 2227890,
Email [email protected]
ABSTRACT
Experimentally determined creep curves for rolling-sliding contact in lubricated
conditions are found to deviate greatly from the standard theory for two body contact.
This paper presents a new model to represent coefficient of adhesion (also known as
traction coefficient) and creep based on experimental data gathered for a range of
railway rail-wheel contact conditions. The model developed is based on a 2D elastic
foundation representation of a twin disc contact. This is used to quantify the creep
curves in a similar manner to existing 3D models of real rail-wheel contacts
undergoing partial slip for a range of lubrication conditions.
The work focuses on very low levels of creep, ranging from zero to 1%, and
lubricants experienced by a rail-wheel contact in service (dry, wet, flange lubricant).
Lubricants used during the simulation of low adhesion conditions for driver training
(soap and water, lignin and water) are also represented. The motivation for the
research is inclusion of creep-traction characteristics in an on-board system being
developed for prediction of low adhesion conditions at the rail-wheel interface based
upon monitoring running conditions prior to brake application.
Keywords: rail-wheel, adhesion, model, 2D, creep, lubrication, railway
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1
Introduction
Rail-wheel contacts operate in a combination of rolling and sliding motions, with
small amounts of creep between the surfaces produced during braking, acceleration,
steering, and through the non-conformal geometry of the rail-wheel contact. In cases
of rail contamination, including autumn leaf debris from surrounding trees, rail-wheel
adhesion can fall to a level insufficient for normal acceleration or braking to be
maintained. In these cases damage to rail and wheels can result from wheel slip or
wheel spin, and there can be safety consequences such as signals passed at danger
(SPADS) [1].
Coefficient of adhesion (CoA, also often referred to as coefficient of traction) is
defined as the ratio of tangential adhesion force (Q) to normal force (P) on the railwheel contact. Guidance notes on low adhesion measurement available from the
Railway Safety and Standards Board (RSSB) [2] state that to sustain normal braking a
CoA of 0.14 or above is required. Beyond warnings at sites where low adhesion
frequently occurs, and seasonal warnings linked to weather and leaf fall, there is
currently little or no advance warning available to the driver that a train has entered a
site of low adhesion until brake application produces poor or insufficient deceleration.
To improve this situation research is underway to develop a low adhesion detection
system [3] capable of predicting low adhesion prior to brake application. A key
component of this is the relationship between CoA and creep at the rail-wheel
interface, which is usually represented by a curve such as Figure 1 (shown in nondimensional form). Critical parameters defining such curves are the creep coefficient
at low slip (the gradient of the adhesion coefficient versus creep curve) and the
saturation level of adhesion coefficient at full slip (i.e. the friction coefficient).
Figure 1 shows theoretical creep curves but these are found to be unrealistic in many
cases. Fletcher and Lewis [4] investigated the link between creep coefficient and
friction coefficient by generating a series of creep curves under closely controlled
conditions for a range of rail-wheel contact contamination conditions. These data
focused on very low levels of creep (0 to 1%) and showed that for contacts lubricated
with flange lubricants, greases or lignin (intended to represent leaf debris) there was
considerable deviation of the creep curve from the form predicted by conventional
rolling-sliding contact theory. To make use of the data from these experiments for low
adhesion detection [3] a model was needed for two-dimensional (line) contact with
partial slip, but it was found that existing models for contaminated contacts focused
primarily on three-dimensional contacts. This paper describes a new model that can
successfully represent these data while maintaining a link to the physical stick and
slip processes within the contact. It is found that just a single additional parameter
beyond those in conventional contact models is needed to represent real creep-force
behaviour.
1.1
Partial slip contact models
When a rail and wheel, or the rail and wheel discs in a twin disc test, are brought
together under a normal load their surfaces deform in both normal and tangential
directions. Within the contact, and depending on the friction coefficient at their
interface, the surfaces may move together with no relative motion, or may move by
different displacements to one another. This divides the contact into regions of ‘stick’
and ‘slip’ respectively, with stick usually happening at the centre of the contact, and
slip happening towards the edges. If the contact is transmitting a shear traction in a
rolling-sliding configuration it is found that the stick region is located at the leading
edge of the contact, and the slip happens at the trailing edge [5] as shown in Figure 2.
The division of the contact into stick and slip regions in this ‘partial slip’ condition
depends on the level of creep relative to the ‘full slip’ case for which transmitted
traction equals the product of normal load and friction coefficient. For dry or
boundary lubrication cases (including water lubrication) the model by Carter [6] and
the elastic foundation model described by Johnson [5] are able to produce very good
two dimensional representations of the experimentally determined creep behaviour in
the transition from pure rolling, through partial slip to the point of full sliding, and
similar models are available for three-dimensional contacts under these conditions
[5,7].
For better lubricated or contaminated contacts, the correlation between the Carter or
elastic foundation models and the experimental data is poor [3]. Factors of importance
for modelling adhesion at rail-wheel contacts in these conditions are summarised by
Polach [8] in development of his three-dimensional contact model. These include
creep dependence of friction coefficient producing different friction levels in the
‘stick’ and ‘slip’ areas of contact, and consideration of contact shear stiffness
reduction caused by an interfacial layer partially separating the rail and wheel.
However, the Polach model is not applicable to two-dimensional contact as it relies a
contact shear stiffness coefficient derived from Kalker’s linear theory [9] which is
inherently three dimensional. Existing two-dimensional models by Carter [6] or the
elastic foundation model by Johnson [5] assume a constant underlying friction
coefficient and do not consider the factors shown by Polach to be important in real
lubricated contacts. Following a summary of experimentally determined creep curve
data for a range of conditions, details are given in Section 3 of an addition to the twodimensional elastic foundation model able to overcome these limitations.
2
Summary of experimental data
This section presents a short summary of the creep curve data generated for a range of
common rail-wheel contact conditions using an updated version of the SUROS rolling
sliding test machine [10]. Full details of the tests and the updates made to the machine
to generate this data are reported elsewhere [3]. This machine is able to maintain
closed loop control of creep between rail and wheel disc samples of 47mm diameter,
each cut out of parent rail and wheel components to ensure the correct materials are
used. The machine was used to perform a programmed variation of creep starting
from pure rolling and increasing to 1% over 60 minutes thereby maintaining a quasistatic creep condition at any moment, and generating an entire creep curve within a
single test. In all cases the test was for a driving wheel, i.e. negative creep using the
conventional definition shown in Equation (1), but all values are presented here as
positive numbers to give conventional creep curve diagrams. The tests used a normal
grade 220 rail of hardness 237HV(100kg) and an R8 wheel of hardness 257HV(100kg) [11].
Creep(%) = 200
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3
A new 2D model to represent creep force
The data generated in the SUROS tests clearly does not all lie on the theoretical creep
curve (Figure 3). Here a model for a modified creep curve is developed capable of
achieving a very close fit to the experimental data for all the conditions tested.
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∋(.∀9%1,∋,%&∀%∗∀∋(.∀∋/!&1,∋,%&∀5.∋+..&∀1∋,∃:∀!&−∀1),9∀∃!&∀5.∀∗%6&−Θ∀Ε.4.45./,&0∀
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∃!&&%∋∀5.∃%4.∀&.0!∋,Φ.<∀0,Φ.1∀2=6!∋,%&∀ΒΡ∆Θ∀
∀
"
"
k K !R %
2k ! R %
c = max $$ a ! a q , 0 '' = max $ a ! a , 0 '
ks K p µ &
3ks µ & ∀
#
#
Β∀Ρ∀∆∀
∀
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!&−∀∋(.∀!/.!∀%∗∀∋(.∀∋/,!&06)!/∀/.0,%&∀5.&.!∋(∀,∋Θ
∀
$ 2ka c 2! R
QA 2c 2 d ! ka K q!
=
a
+
x
( )& =
#
P
P dx " h
a3
%
∀
∀
∀
∀
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∀
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∀
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Ζ(.∀&%/4!)∀∗%/∃.∀>∀,1∀0,Φ.&∀5Η∀.=6!∋,%&∀ΒϑΟ∆Θ∀
,
2K p a 3
P=
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∀
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∀
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[&∀ ∋(.∀ 1),99,&0∀ /.0,%&<∀ ∋(.∀ 1(.!/∀ ∗%/∃.∀ ∋/!&14,∋∋.−∀ ,1∀ 0,Φ.&∀ 5Η∀ 2=6!∋,%&∀ Βϑϑ∆∀
,,
+(,∃(∀+!1∀0.&./!∋.−∀5Η∀,&∋.0/!∋,&0∀2=6!∋,%&∀ΒΥ∆∀5.∋+..&∀∋(.∀),4,∋1∀)(∗∀∀∋%∀∀Θ
,
QS 1
=
P P
"
a!2c
!a
(
3
)
3
ks K p µ 2 2
3 µ ks ( a ! c) 1 µ ks a ! ( 2c ! a )
(a ! x )dx = 2 a ! 4
2Rh
a3
∀
Β∀ϑϑ∀∆,
∀
Ζ!:,&0∀ ∋(.∀ 1∋,∃:∀ !&−∀ 1),9∀ /.0,%&1∀ ∋%0.∋(./<∀ 2=6!∋,%&∀ ΒϑΚ∆∀ 0,Φ.1∀ ∋(.∀ ∋%∋!)∀ ∃/..9∀
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∀
(
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+
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a3 #
2 ######
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4 "######
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$ !
$∀
Adhesion region
∀
Slip region
∀
Β∀ϑΚ∀∆∀
4
Application of two dimensional model to experimental data
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5..&∀9/.Φ,%61)Η∀−.∗,&.−∀∗/%4∀∋(.∀1!∋6/!∋,%&∀).Φ.)∀%∗∀!−(.1,%&∀∃%.∗∗,∃,.&∋∀!∋∀∗6))∀
1),9∀ΞΝΨ∀!&−∀!/.∀1644!/,1.−∀,&∀Ζ!5).∀ϑΘ∀Ζ(.∀−!∋!∀∗,∋∋,&0∀+!1∀∃!//,.−∀%6∋∀61,&0∀∋(.∀
Χ∗,∋8∀ ∗6&∃∋,%&∀ +,∋(,&∀ δ&69)%∋∀ ΞϑΝΨ∀ +(,∃(∀ ,49).4.&∋1∀ !∀ &%&),&.!/∀ ).!1∋Γ1=6!/.1∀
Π!/=6!/−∋Γε.Φ.&5./0∀!)0%/,∋(4Θ∀[∋∀+!1∀∗%6&−∀∋(!∋∀∗,∋∋,&0∀61,&0∀β61∋∀9!/!4.∋./∀#∀∀
+!1∀1%∀16∃∃.11∗6)∀∋(!∋∀∗6/∋(./∀∗,∋∋,&0∀5Η∀Φ!/,!∋,%&∀%∗+#∃<∀+(,∃(∀,1∀4%1∋∀/.).Φ!&∋∀∋%∀
∃%&∋!∃∋1∀%9./!∋,&0∀!∋∀∃/..9∀5.Η%&−∀∗6))∀1),9<∀+!1∀&%∋∀96/16.−Θ∀Ζ(.∀−!∋!∀61.−∀(./.∀
−%∀ &%∋∀ .&∋./∀ ∋(!∋∀ /.0,%&<∀ (%+.Φ./<∀ ,&∃)61,%&∀ ,&∀ ∋(.∀ 4%−.)∀ %∗∀ ∋(.∀ #∃∀ 9!/!4.∋./∀
−%.1∀%∗∗./∀∋(.∀%99%/∋6&,∋Η∀∋%∀4%−.)∀∗%/∀∋+%∀−,4.&1,%&!)∀∃%&∋!∃∋1∀∋(.∀−.∃),&.∀,&∀
!−(.1,%&∀ ∃%.∗∗,∃,.&∋∀ !∋∀ (,0(∀ ∃/..9∀ ∋(!∋∀ +!1∀ /.9/.1.&∋.−∀ ∗%/∀ ∋(/..Γ−,4.&1,%&!)∀
∃%&∋!∃∋1∀5Η∀>%)!∃(∀ΞΡΨ<∀!&−∀∋(,1∀,1∀∋(.∀165β.∃∋∀%∗∀∗6/∋(./∀/.1.!/∃(Θ∀
∀
Ζ!5).∀ Κ∀ 1(%+1∀ ∋(.∀ 9!/!4.∋./∀ Φ!)6.1∀ 9/%−6∃.−∀ 5Η∀ ∗,∋∋,&0∀ ∋(.∀ &.+∀ Κ⊥∀ 4%−.)∀ ∋%∀
.!∃(∀ %∗∀ ∋(.∀ .Λ9./,4.&∋!)∀ −!∋!1.∋1Θ∀ Μ,06/.∀ Τ∀ 1(%+1∀ ∋(.∀ /.16)∋1∀ ∗%/∀ ∃!1.1∀ %∗∀ %,)<∀
1%),−∀)65/,∃!∋.−<∀!&−∀−/Η∀∃%&∋!∃∋1Θ∀[&∀.!∃(∀9)%∋∀∋(.∀%/,0,&!)∀4%−.)∀Β,Θ.Θ∀#∀φϑ∀!&−∀
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Φ!)6.1∀ %∗∀ #∀∀ 1(%+&∀ ,&∀ Ζ!5).∀ ΚΘ∀ [∋∀ ∃!&∀ 5.∀ 1..&∀ ∗/%4∀ ∋(.∀ ∗,06/.1∀ ∋(!∋∀ ∋(.∀ &.+∀ Κ⊥∀
4%−.)∀,1∀!5).∀∋%∀9/%−6∃.∀46∃(∀5.∋∋./∀!0/..4.&∋∀+,∋(∀∋(.∀.Λ9./,4.&∋!)∀−!∋!Θ∀Ζ(.∀
/.∗,&.4.&∋∀,1∀4,&%/∀,&∀∋(.∀∃!1.∀%∗∀−/Η∀∃%&∋!∃∋<∀∗%/∀+(,∃(∀∋(./.∀,1∀&%∀∋(,/−∀5%−Η∀
9/.1.&∋∀ ,&∀ ∋(.∀ ∃%&∋!∃∋<∀ !&−∀ ∗/,∃∋,%&∀ 5.(!Φ,%6/∀ ,1∀ ∃)%1.)Η∀ !99/%Λ,4!∋.−∀ 5Η∀ ∋(.∀
1∋!&−!/−∀ 4%−.)∀ ΒΜ,06/.∀ Ν∆Θ∀ [&∀ ∋(.∀ ∃!1.∀ %∗∀ +.))∀ )65/,∃!∋.−∀ ∃%&∋!∃∋1∀ ∋(.∀ %/,0,&!)∀
4%−.)∀ 9/%−6∃.−∀ !∀ 9%%/∀ /.9/.1.&∋!∋,%&∀ %∗∀ ∋(.∀ ∃/..9∀ ∃6/Φ.<∀ +(./.!1∀ ∋(.∀ &.+∀∀
4%−.)∀ ,1∀ 46∃(∀ 5.∋∋./∀ !5).∀ ∋%∀ ∃!9∋6/.∀ ∋(.∀ 1)%9.∀ !∋∀ )%+∀ ∃/..9∀ Φ!)6.1∀ !&−∀ ∋(.∀
0/!−6!)∀∋/!&1,∋,%&∀∋%∀∗6))∀1),9Θ∀γ6/Φ.1∀∗%/∀%∋(./∀)65/,∃!&∋1∀1(%+.−∀1,4,)!/)Η∀Φ./Η∀
0%%−∀!0/..4.&∋∀5.∋+..&∀∋(.∀&.+∀4%−.)∀!&−∀∋(.∀.Λ9./,4.&∋!)∀−!∋!Θ∀∀
∀
Μ,06/.∀ Υ∀ 1(%+1∀ ∋(.∀ /.16)∋1∀ %∗∀ !99)Η,&0∀ ∋(.∀ 4%−.)∀ ∋%∀ ∃/..9∀ ∃6/Φ.1∀ !∋∀ !∀ /!&0.∀ %∗∀
∃%&∋!∃∋∀9/.116/.1<∀61,&0∀∋(.∀∗/,∃∋,%&∀∃%.∗∗,∃,.&∋∀Φ!)6.1∀∗/%4∀Ζ!5).∀ϑ<∀∋(.∀#∀∀Φ!)6.1∀
,&∀Ζ!5).∀Κ∀!&−∀∋!:,&0∀!∃∃%6&∋∀%∗∀∋(.∀∃(!&0.∀,&∀∃%&∋!∃∋∀9!∋∃(∀(!)∗Γ+,−∋(∀61,&0∀∋(.∀
Φ!)6.1∀ ∗%/∀ ∀∀ 0,Φ.&∀ ,&∀ ≅.∃∋,%&∀ ΚΘ∀ Ζ(.∀ −%∋∋.−∀ ),&.1∀ ,&∀ ∋(.∀ ∗,06/.∀ ,&−,∃!∋.∀ ∋(.∀
9/.−,∃∋,%&1∀ %∗∀ ∋(.∀ 6&4%−,∗,.−∀ .)!1∋,∃∀ ∗%6&−!∋,%&∀ 4%−.)∀ ∗%/∀ ∋(.1.∀ ∃%&−,∋,%&1<∀
1(%+,&0∀ Φ./Η∀ 9%%/∀ !0/..4.&∋∀ ,&∀ ∋(.∀ 9!/∋,!)∀ 1),9∀ /.0,%&∀ !∋∀ !))∀ ∋(.∀ ∃%&∋!∃∋∀
9/.116/.1Θ∀ η1,&0∀ ∋(.∀ &.+∀ 4%−.)∀ ∋(.∀ ∃6/Φ.1∀ 46∃(∀ 5.∋∋./∀ /.9/.1.&∋∀ ∋(.∀
.Λ9./,4.&∋!)∀−!∋!<∀!)∋(%60(∀∋(.Η∀1),0(∋)Η∀%Φ./Γ9/.−,∃∋∀∋(.∀!−(.1,%&∀∃%.∗∗,∃,.&∋∀,&∀
∋(.∀∃/..9∀/!&0.∀ΟΘΤΓΟΘΣι∀∗%/∀∃%&∋!∃∋∀9/.116/.1∀%∗∀ϑΟΟΟ∀!&−∀ϑΝΟΟΠ>!Θ∀∀
∀
4.1
Physical meaning
[&∀!−−,∋,%&∀∋%∀5.∋∋./∀/.9/.1.&∋,&0∀∋(.∀.Λ9./,4.&∋!)∀−!∋!<∀∋(.∀&.+∀4%−.)∀/.∋!,&1∀
∃).!/∀ ∃%&∋/,56∋,%&1∀ ∋%∀ ∃%&∋!∃∋∀ 1(.!/∀ ∗%/∃.∀ ∗/%4∀ ∋(.∀ 1∋,∃:∀ !&−∀ 1),9∀ !/.!1∀ %∗∀ ∋(.∀
∃%&∋!∃∋<∀ +(,∃(∀ !))%+1∀ 1%4.∀ ,&1,0(∋∀ ,&∋%∀ ∋(.∀ 9(Η1,∃!)∀ 4.!&,&0∀ %∗∀ ∋(.∀ /.−6∃∋,%&∀
∗!∃∋%/1Θ∀∀Μ,06/.∀Σ∀1(%+1∀!∀∃%49!/,1%&∀%∗∀∋(.∀%/,0,&!)∀.)!1∋,∃∀∗%6&−!∋,%&∀!&−∀&.+∀
Κ⊥∀4%−.)1<∀,&−,∃!∋,&0∀(%+∀∋(.∀1(.!/∀∗%/∃.∀,&∀∋(.∀!−(.1,%&∀Β1∋,∃:∆∀!&−∀1),9∀!/.!1∀
%∗∀∃%&∋!∃∋∀∃%&∋/,56∋.1∀∋%∀∋(.∀∋%∋!)∀1(.!/∀∗%/∃.∀%Φ./∀!∀/!&0.∀%∗∀)%+∀∃/..9∀Φ!)6.1Θ∀
Ζ(.∀∗,06/.∀,1∀5!1.−∀%&∀∋(.∀%,)∀)65/,∃!∋.−∀∋.1∋∀ϕϑ∀∗%/∀+(,∃(∀∋(.∀%/,0,&!)∀4%−.)∀+!1∀
!∀Φ./Η∀9%%/∀/.9/.1.&∋!∋,%&∀%∗∀∋(.∀−!∋!<∀!&−∀∋(.∀&.+∀4%−.)∀+,∋(∀#∀φΟΘϑαΝ∀46∃(∀
5.∋∋./∀/.9/.1.&∋1∀∋(.∀−!∋!Θ∀[∋∀∃!&∀5.∀1..&∀∋(!∋∀,&∃)61,%&∀%∗∀∋(.∀9!/!4.∋./+#∀∀(!1∀
∋(.∀ .∗∗.∃∋∀ %∗∀ ∀ Χ1∋/.∋∃(,&08∀ ∋(.∀ 9!/∋,!)∀ 1),9∀ !/.!∀ %∗∀ ∋(.∀ ∃/..9∀ ∃6/Φ.∀ %Φ./∀ !∀ 46∃(∀
)!/0./∀ /!&0.∀ %∗∀ ∃/..9∀ /!∋,%<∀ ∋(!∋∀ ,1<∀ ∋(.∀ 9(Η1,∃!)∀ 4.!&,&0∀ %∗∀ ∋(.∀ 9!/!4.∋./∀ ,1∀ !∀
−.)!Η∀,&∀∋(.∀∃/..9∀!∋∀+(,∃(∀∗6))∀1),9∀,1∀!∃(,.Φ.−Θ∀Ζ(,1∀−.)!Η∀,1∀9!/∋,∃6)!/)Η∀/.).Φ!&∋∀
∋%∀+.))∀)65/,∃!∋.−∀∃!1.1<∀∗%/∀+(,∃(∀∋(.∀%/,0,&!)∀4%−.)∀).!1∋∀+.))∀/.9/.1.&∋.−∀∋(.∀
.Λ9./,4.&∋!)∀−!∋!Θ∀∀
∀
5
Conclusions
A two-dimensional model has been developed to represent the real traction-creep
behaviour of well lubricated rolling-sliding contacts. The model is able to accurately
represent creep cuve data gathered from experiments on a twin-disc testing machine
under lubrication conditions characteristic of the railway rail-wheel contact. The data
for which the model has been demonstrated focus on very low levels of creep, ranging
from zero to 1%, for lubrication conditions experienced by a rail-wheel contact in
service (dry, wet, flange lubricant). Those used during the simulation of low adhesion
conditions for driver training (soap and water, lignin and water) were also
investigated.
The model is based on an elastic foundation representation of the contact, and is able
to represent real adhesion behaviour for a two-dimensional contact in the same way
that an earlier model by Polach [8] achieves this for three-dimensional contacts. This
includes the effect of creep dependent friction coefficient (for example due to
temperature differences with slip) and contact shear stiffness reduction caused by an
interfacial layer between the rail and wheel steel surfaces. The current model is
independent of the coefficients derived from Kalker’s linear theory on which the
Polach model is based, and which prevent its direct application to two-dimensional
contacts.
The motivation for the research was inclusion of adhesion-creep characteristics in an
on-board system being developed for prediction of low adhesion conditions at the railwheel interface, based upon monitoring running conditions prior to brake application.
The model developed provides a simple way to represent real adhesion-creep
behaviour, and to link differences in creep coefficient (initial slope of the adhesion
coefficient-creep curve) to friction coefficient (saturation level of adhesion coefficient
at full slip). Use in a predictive capacity to map contact forces measured at low levels
of slip to the maximum available adhesion will support automated detection and
warning of low adhesion before braking is attempted.
6
Acknowledgements
The author is grateful to the Railway Safety and Standards Board for their
sponsorship of this research, and for the collaboration with Professor Roger Goodall
and Dr Chris Ward at Loughborough University that has made the research possible.
∀
7
References
1.
Poole, W, An Overview of Low Adhesion Factors for ERTMS, Project T080 final report, July
2003, Rail Safety and Standards Board, Block 2, Angel Square, 1 Torrens Street, London
EC1V 1NY
2.
GM/GN2642 Guidance on Wheel/ Rail Low Adhesion Measurement, Issue One: February
2008, Rail Safety and Standards Board, Block 2, Angel Square, 1 Torrens Street, London
EC1V 1NY
3.
Onboard detection of low adhesion, Railway Safety and Standards Board project T959,
Research and Development E-Newsletter, Issue 69, January 2011, www.rssb.co.uk.
4.
Fletcher, DI, Lewis, S, Creep curve measurement using a continuously variable creep twin
disc machine, submitted to Wear, 2012.
5.
KL Johnson, Contact Mechanics, Cambridge University Press, 1987, Cambridge, UK
6.
Carter, FW, On the Action of a Locomotive Driving Wheel, Proceedings of the Royal Society
of London. Series A, 112, 760 (1926),151-157
7.
Haines, DJ, Ollerton, E, Contact stress distribution on elliptical contact surfaces subject to
radial and tangential forces, Proceedings of the Institution of Mechanical Engineers, 177, 95
(1963) 261-265
8.
O. Polach, Creep forces in simulations of traction vehicles running on adhesion limit, Wear
258 (2005) 992–1000
9.
J.J. Kalker, On the rolling contact of two elastic bodies in the presence of dry friction, Thesis,
Delft, 1967.
10. Fletcher D. I. and Beynon, J. H. Development of a machine for closely controlled rolling
contact fatigue and wear testing. J. Test. Eval., 2000, 28(4), 267–275.
11. Carroll, RI, Beynon, JH, Rolling contact fatigue of white etching layer: Part 1: Crack
morphology, Wear, Volume 262, Issues 9-10, 2007, 1253-1266
12. Engineering Science Data Unit Item 78035, Contact phenomena I: stresses, deflections and
contact dimensions for normally-loaded unlubricated elastic components, 1995, ESDU
International, London.
13. Williams, T, Kelley, C, Gnuplot 4.4, An Interactive Plotting Program, manual available from
www.gnuplot.info.
∀
Tables
Table 1. Lubrication conditions, test codes, friction coefficients (saturation adhesion
coefficient) and creep coefficients (initial force-creep gradients) for each
test
Lubricant type∀
]!∋./∀
≅%),−∀)65/,∃!&∋∀
≅%),−∀)65/,∃!&∋∀
≅%),−∀)65/,∃!&∋∀
≅%!9∀!&−∀+!∋./∀
ϕ,)∀
Ζ/!∃:∀0/.!1.∀Ζδ∀
Ζ/!∃:∀0/.!1.∀γδ∀
ε,0&,&∀!&−∀+!∋./∀
⊥/Η∀
∀
γ%&∋!∃∋∀
9/.116/.<∀Π>!∀
ϑΟΟΟ∀
ϑΟΟΟ∀
ΡΟΟ∀
ϑΝΟΟ∀
ϑΟΟΟ∀
ϑΟΟΟ∀
ϑΟΟΟ∀
ϑΟΟΟ∀
ϑΟΟΟ∀
ϑΟΟΟ∀
Ζ.1∋∀∃%−.∀
]ϑ∀
≅εϑ∀
≅εΚ∀
≅εΝ∀
≅]ϑ∀
ϕϑ∀
Ζδϑ∀
γδϑ∀
εϑ∀
⊥Ν∀
Μ/,∃∋,%&∀
∃%.∗∗,∃,.&∋<∀µ∀∀
ΟΘΚΤ∀
ΟΘϑΚ∀
ΟΘϑΣ∀
ΟΘϑΟΝ∀
ΟΘΟΥΥ∀
ΟΘΟΥΥ∀
ΟΘΟα∀
ΟΘΟΚΥ∀
ΟΘΟΡ∀
ΟΘΥΡ∀
γ/..9∀
∃%.∗∗,∃,.&∋∀∀
ΟΘ_∀
ΟΘΤΥ∀
ΟΘΤ∀
ΟΘΤ∀
ΟΘΚ∀
ΟΘΚΤ∀
ΟΘΚΣ∀
ΟΘΟΡ∀
ΟΘΚ_∀
ΟΘ_Υ∀
Table 2. Parameter values used in new 2D model to achieve fits to the experimental
data. The value of ks was taken to be 1 in all cases.
Ζ.1∋∀
]ϑ∀
≅εϑ∀
≅εΚ∀
≅εΝ∀
≅]ϑ∀
ϕϑ∀
Ζδϑ∀
γδϑ∀
εϑ∀
⊥Ν∀
∀
∀
#∀ +
ΟΘΡΟΥ∀
ΟΘΤΣΚ∀
ΟΘΝΟΣ∀
ΟΘΤΣΤ∀
ΟΘΚΝα∀
ΟΘϑαΝ∀
ΟΘΚΝΝ∀
ΟΘΟΤαΥ∀
ΟΘϑΣΡ∀
ΟΘΡΟΡ∀
Figure Captions
Figure 1.
Comparison of creep curves form the Carter (Hertzian) and elastic
foundation models
Figure 2.
Distribution of normal pressure and tangential traction across a
partially slipping contact.
Figure 3.
Creep curves for all the lubrication conditions examined, normalised
axes. All at 1000MPa maximum Hertzian contact pressure.
Figure 4.
Creep curves, with experimental data alongside curves from the
original and new 2D contact models. (a) Oil lubricated contact, test
O1. (b) Data for solid lubricant from test SL1. (c) Dry test, D3.
Figure 5.
Creep curves for tests on solid flange lubricant at a range of contact
pressures. Experimental data (points), predictions of original model
(dotted lines) and new model (solid lines).
Figure 6.
Creep curves for the original and new 2D model, plotted for oil
lubricated test O1. For each model the total creep force curve is shown
alongside the contributions from the stick (adhesion) and slip areas of
the contact. Parameters for the model are taken from Table 1 and Table
2.
∀
∀
∀
∀
∀
Μ,06/.∀ϑ∀
∀
∀
∀
∀
∀
Μ,06/.∀Κ∀
∀
∀
∀
Μ,06/.∀Ν∀
∀
∀
∀
∀
∀
Μ,06/.∀Τ!∀
∀
∀
∀
Μ,06/.∀Τ5∀
∀
∀
∀
∀
∀
Μ,06/.∀Τ∃∀
∀
∀
∀
Μ,06/.∀Υ∀
∀
∀
∀
∀
∀
Μ,06/.∀ΣΘ∀